Cubic Quartets With a,b,c < 1000

The table below is seriously impressive in one respect: the program that generated it contained three
nested loops each going to 1000. Counting the power calculations and primality
checks, the program easily ran to several billion calculations. It ran in less
than ten minutes on a not very state of the art desktop computer. Considering
that I can remember when computer speeds were measured in kilohertz and
large files in kilobytes, I find this impressive. And yes, the listing did stop
at a = 846. In fact, when a is small, the quartets often stop with b and c
values well short of 1000.